Optimization Method For The Design of Axial Hydraulic Turbines
Optimization Method For The Design of Axial Hydraulic Turbines
Optimization Method For The Design of Axial Hydraulic Turbines
A Lipej
Turboins† titut, Rovs† nikova 7, 1210 Ljubljana, Slovenia
Abstract: Computational uid dynamics (CFD) is becoming an increasingly reliable tool for the design of
water turbines. Using different CFD codes, it is possible to nd out and compare criteria for classifying
runner blade geometry regarding the strengths of their characteristics. The nal decision of runner geometry,
with demanding energetic and cavitation characteristics, always remains for the design engineer. To reach the
nal result, the engineer has to compare the ow analysis results of a great number of different geometries.
To replace a part of this work, an optimization algorithm has been developed. This optimization procedure
helps to check many more geometries with less human work. In this paper, a multiobjective genetic algorithm
for the design of axial runners is presented. For the ow analysis, the CFX-TASC ow code, with a standard
k–e turbulence model, has been used, because the code enables calculation within a rotating frame of
reference. For design of the initial geometry, within the optimization procedure, a special program has been
developed, which makes it possible to start the optimization procedure with a relatively high level of
ef ciency and transforms prescribed genetic parameters to the runner geometry.
Keywords: water turbines, ef ciency, cascade, numerical method, genetic algorithm
optimization with two criteria has been used, and the results been made. Accuracy of the numerical results is important
have been compared with measurements. The key points of in order to obtain a reliable objective function for the design
the genetic algorithms are the operators used for selection optimization procedure. The last part presents a multiobjec-
and reproduction which have a great in uence on the tive, genetic optimization algorithm, coupled with the
robustness and the ef ciency of the algorithm. design program and numerical ow analysis.
The main purpose of the ow analysis is to predict ow
properties and losses and to obtain reliable criteria for
2 DESIGN OF AXIAL RUNNERS
classifying runners with good or bad characteristics. From
the pressure distribution on the runner blades, the torque on
the shaft was calculated. In the runner, most of the differ- Development of the new runner blade shapes can be done
ence in total pressure is converted to runner work, while a using different direct or inverse design methods [2]. The
small part presents ow energy losses. The turbine ef - development of a new runner starts with the de nition of
ciency can be calculated by the following parameters:
(a) nominal and maximal head,
X Mo
Mˆ pi bxi Ayi ¡ yi Axi c and Zˆ (b) nominal power,
i
rQ DE (c) predicted ef ciency,
(d) predicted cavitation coef cient.
where M is the torque on the shaft, p is the static pressure, x
The simpli cation of the ow analysis is based on the
and y are components of distance from the axis, Ax and Ay
fundamental features of the ow in axial turbines. Inside
are area projections, index i represents all elements on the
the channel, in front of the runner and downstream of
blade surface, o is the angular velocity and DE is obtained
the runner, the ow is axisymmetric, especially for the
from the difference between total pressure, pt at the domain
optimal operating point. Inside the runner the stream
inlet and outlet
surfaces are cylindrical [3], which is the consequence of
Á ! the axisymmetric ow. Taking into account the hypothesis
1 X X
DE ˆ ptj Qj ¡ ptj Qj of cylindrical stream surfaces, the three-dimensional ow
rQ j(inlet) j(outlet) can be analysed as two-dimensional ow through the axial
runners (Fig. 2), where c is absolute velocity, w is relative
where Q is the owrate and index j represents all the velocity and u represents the rotational speed.
elements on the inlet and outlet cross-sections.
The design procedure is based on the use of theoretical
hydrodynamic cascade characteristics for potential ow and
empirical ow corrections on account of viscosity effects.
The resulting cascade and pro le camber means the optimal
solution for given initial parameters. The design program is
used rst of all to start the optimization of the runner blade
geometry with relative high ef ciency, which means a
signi cant reduction in the CPU time.
In this paper the theory of axial turbines and the design
procedure for axial runners are presented rst. The second
part deals with three-dimensional turbulent ow analysis of
different speci c speed axial runners for various operating
regimes. A comparison of energetic and kinematic charac- Fig. 2 Velocity traingles at the inlet (index 1) and outlet
teristics between numerical and experimental results has (index 2) of the runner
Proc. Instn Mech. Engrs Vol. 218 Part A: J. Power and Energy A02803 # IMechE 2004
OPTIMIZATION METHOD FOR THE DESIGN OF AXIAL HYDRAULIC TURBINES 45
where cu2 is the circumferential component of absolute velocity Fig. 4 Comparison of (a) calculation and (b) measurement of
at the outlet and Dcu is the difference between the inlet and relative ef ciency for three different runners (r1, r2, r3)
outlet circumferential components of absolute velocity. The in the same operating regime
value of the coef cient Ku2 is zero, or varies linearly from ¡0.4
near the hub to 0.4 on the peripheral cylindrical section.
A computer program for the design of the optimal shape of carried out on the model of a four-bladed Kaplan turbine in
axial runner [4] also prepares the les necessary for the grid
the Turboins† titut.
generationprocedure with the CFX-TASCgrid code [5]. Before
According to the experimental results, behind the hub
the solver can start the calculation,some data about the number
there is an area of small velocities, but in the case of
of elements and density of the elements have to be determined. numerical analysis this area is smaller. This is the conse-
quence of the horizontal pier in the draft tube, which was not
3 NUMERICAL FLOW ANALYSIS taken into account in the numerical analysis.
The numerically and experimentally obtained ef ciencies
The numerical grids for all the calculations through the for three different runners are also presented in Fig. 4. The com-
runner consist of 32 000 nodes. The computational domain parison shows that using the numerical results for classifying
is reduced to the ow eld between two runner blades, a part the designed runners is successful. The ef ciency measure-
in front of the runner and the rst part of the diffuser. The ment can be predicted with up to 0.3 per cent accuracy;
computational domain between the two runner blades can however, the accuracy of numerically obtained ef ciency is
be analysed because periodical boundary conditions can be about 2 per cent.
used. The ow analysis in the axial runner can be started
when the boundary condition is prepared.
4 GENETIC ALGORITHM FOR
Comparing the numerical results with experimental ones
MULTIOBJECTIVE OPTIMIZATION
is a good test for the accuracy of the numerical method. On
the test rig installed in the Turboins† titut in Ljubljana, many
The basic idea of the optimization procedure is to nd the
energetic and cavitation measurements were carried out,
shape of the axial runner (Figs. 1 and 10) that gives the
including some measurements of the ow kinematics at
required energetic and cavitation characteristics. After de n-
the outlet of the runner (Fig. 3). These measurements were
ing the design operating point, all the necessary parameters
have to be determined. Firstly, cylindrical sections are
selected. Then, the parameters are given as a function of
the relative diameter:
(a) the meridional velocity coef cient (two variables),
(b) the outlet vortex coef cient (two variables),
(c) the chord–pitch ratio (two variables),
(d) the relative pro le maximum thickness (two variables),
(e) the camber position (two variables).
All the parameters can be either linear or a higher-order
function of the relative diameter. The optimization has
therefore been limited to the treatment of ten variables.
For the genetic algorithm, the binary string (Table 1) for
the runner shape has to be determined. This string consists
Fig. 3 Comparison of experimentally and numerically of prescribed parameters that can be converted to the runner
obtained velocity components at the outlet of the geometry using the design program so that the runner is
runner: va , vr , vu —numerical results; va m , vr m , vu m — prepared for evaluation [6]. The shape of the runner alone is
experimental results not enough to provide all the characteristics required, and so
A02803 # IMechE 2004 Proc. Instn Mech. Engrs Vol. 218 Part A: J. Power and Energy
46 A LIPEJ
Fig. 5 Ef ciency distribution for runners in all generations: (a) small changes in initial parameters;
(b) non-linear parameter distribution
Proc. Instn Mech. Engrs Vol. 218 Part A: J. Power and Energy A02803 # IMechE 2004
OPTIMIZATION METHOD FOR THE DESIGN OF AXIAL HYDRAULIC TURBINES 47
Fig. 6 (a) Distribution of b and (b) distribution of f/l for all runners in the rst generation
ef ciency is in the rst generation; for later generations the design parameters. In Fig. 7b the chord–pitch ratio is
results are almost constant. The possibility of obtaining an presented. In the optimization procedure each parameter
optimal solution in such a case is very small, because the has a different in uence on the ef ciency, which is why
in uence of initial conditions is dominant. In further analy- some parameters remain equal through most generations.
sis the limits of the initial parameters are different, and also This can be seen in Fig. 7b, where the function l/t is very
a non-linear distribution of parameters can be used. similar up to the eighth generation and only in the last few
The distribution of ef ciency (Fig. 5b) and pressure generations do signi cant differences occur. From the
coef cient shows high uctuations in the rst few genera- difference in l/t in the last generations, some conclusions
tions. In this case (b) the limits of initial parameters are about the number of generations can be obtained. To obtain the
more distant than in case (a), and with a non-linear para- optimal solution, the number of generations has to be higher.
meter distribution. This can also be shown with some During the optimization procedure, the genotypes are
geometrical characteristics of the runner blade. different. In the rst generation, when the parameters are
In Fig. 6 two main parameters are shown: the skewness of chosen randomly, the difference from average values is
the blade, b, and the relative curvature of the pro les, f /l, for quite large. In the following generations the situation
all runners in the rst generation. A difference in uctua- improves. In the last generation, the average value for all
tions between the rst and last generation is obtained in all ten parameters is almost constant.
optimization cases. As a second criterion, the relative pressure number is
In all cases of the optimization procedure there are ten taken into account (Figs 8 and 9). The relative pressure
individuals in each generation. To keep the maximum CPU number is de ned by the equation
time for one optimization procedure within about 2 days, the
number of generations can vary from 16 to 20. cnum
Kc ˆ 1 ¡ 1 ¡
The distribution of the b angle for all runners in the last co
generation is so uniform that the difference in Fig. 7a is
almost invisible. The design procedure of the runner can where cnum is a numerically obtained pressure number
also be presented by diagrams showing the change in the and co is the prescribed optimal value of the pressure
Fig. 7 (a) Distribution of b for all runners in the last generation and (b) distribution of the chord–pitch ratio
for six generations
A02803 # IMechE 2004 Proc. Instn Mech. Engrs Vol. 218 Part A: J. Power and Energy
48 A LIPEJ
Fig. 8 (a) Distribution of ef ciency and (b) distribution of relative pressure number for all individuals
Fig. 9 (a) Distribution of average ef ciency and (b) distribution of relative pressure number for all generations
Fig. 10 (a) Comparison of shapes for all pro les near the hub and at maximum radius in the rst (a), tenth (b)
and last generation (c), shown as an evolution process
Proc. Instn Mech. Engrs Vol. 218 Part A: J. Power and Energy A02803 # IMechE 2004
OPTIMIZATION METHOD FOR THE DESIGN OF AXIAL HYDRAULIC TURBINES 49
Fig. 11 Pressure distribution for the designed runner from the sixth (a) and last generation (b)
number. The relative pressure number is expressed by the The design program, together with experience gained,
equation makes it possible to start the optimization procedure for the
runner with a high level of ef ciency. It is developed
E especially for use with numerical ow analysis and an
co ˆ
(p2 =2)n2 D2 optimization algorithm. The design program for axial
runners is responsible for transformation from genotype to
As a result of the optimization procedure, the comparison geometry data [9].
of geometric parameters of runner blades through all To ascertain which geometry has good or bad char-
generations is presented. In Fig. 10 the shapes of two acteristics, numerical ow analysis inside the runner is
pro les for each individual from different generations are used, and one of the possible presentations of the results
presented. Each runner is presented by the pro le near is the pressure distribution (Fig. 11). From a great number
the hub and the pro le at the maximum radius (Fig. 1). In of analysed runners with different speci c speeds, data for
the rst half of the generation, all of the parameters are the inlet meridional velocity and the outlet vortex have
highly variable. After the rst half of the generation, the been obtained. The calculated values can be used instead
blade angles do not change any further, but the para- of the theoretical ones necessary for runner blade design.
meters responsible for length and maximum curvature still Using the calculated parameters, the initial results for the
vary. Figure 10 presents all geometries for each genera- energetic characteristics of the runners are better than
tion, but the difference between cases a to c is that in the when theoretical values are used. The nal results for
rst generation (a) the geometries are different, in the the ef ciency of the initial runner and the optimized
tenth generation some runners are already very similar (b) runner, obtained by model measurements, are presented
and in the last generation (c) the differences are so small in Table 2.
that pro les overlap each other. Also, the pressure distri-
bution for the geometry from different generations is
compared. 6 CONCLUSIONS
Table 2 Difference of ef ciency for This paper describes the complete optimization procedure
initial and nal runner for all for the design of axial runners. The procedure consists of a
operating regimes design program, numerical ow analysis and a multiobjec-
tive genetic algorithm. The inlet boundary condition can be
Q/Qopt (—) DZ (%)
obtained from a previous calculation of a tandem cascade.
0,39 1,7 On the basis of the numerical results, the genetic algorithm
0,58 1,3
0,81 0,8 makes new generations using operators for selection and
1 0,6 reproduction. In spite of the fact that three-dimensional
1,23 0,55 numerical ow analysis is very time consuming, using
1,61 0,5
the program for runner blade design enables the whole
A02803 # IMechE 2004 Proc. Instn Mech. Engrs Vol. 218 Part A: J. Power and Energy
50 A LIPEJ
Proc. Instn Mech. Engrs Vol. 218 Part A: J. Power and Energy A02803 # IMechE 2004